#123710

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#397
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(5,6,7,8)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,32)(24,35)(26,34)(28,33)(29,31)(30,38)(36,37)(39,40);(1,2)(3,4)(9,11)(10,16,20,31)(12,22)(13,23)(14,29)(15,24)(17,32,34,30)(18,21,19,35)(25,36,33,40)(26,37)(27,39)(28,38);(2,4)(9,12,13,26,15,28,14,27)(10,17,18,33,20,34,19,25)(11,22,23,37,24,38,29,39)(16,30,21,36,31,32,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30906);